In various businesses and industries, it is desirable to be able to clean small parts. By way of example, is often desired to clean small automotive parts, to remove grease, oil, paint, dirt, or other contaminants. Such small parts are frequently cleaned in an abrasive blast cleaning cabinet, sometimes referred to as a bead blaster. A typical abrasive blast cleaning cabinet is illustrated in U.S. Pat. No. 5,453,039 and is there described as including an abrasive collector bin and a cover member having a viewing window and arm apertures, with gloves attached to the arm apertures so that a persons hands can be inserted into the gloves to grip small parts within the abrasive blast cleaning cabinet. However, use of the abrasive blast cleaning cabinet in this manner requires the person to hold the small parts as they are being cleaned by blasting with abrasive material, such as sand. This inhibits the abrasive material from reaching all locations on the small parts and further presents the hazard of the person inadvertently dropping the small parts, with the result that the small parts become lost within used abrasive material that has collected on the bottom of the cleaning cabinet. To avoid this, those small parts having a suitable opening might be strung on a wire which is suspended in the abrasive blast cleaning cabinet. However, this is time consuming and tedious. Further, many small parts do not have an opening for passage of such a wire. U.S. Pat. No. 5,453,039 proposes to overcome these problems by providing a cleaning basket having a cylindrical main body portion and a frusto-conical neck portion that fits within the upper end of the main body portion, with a screen closing the lower end of the main body portion. The neck portion lower end cooperates with the upper end of the main body portion to define an area that intercepts small parts that are agitated within the basket, reducing the likelihood that those small parts will exit the basket through the open upper end. The abrasive material passes through screen on the lower end of the abrasive cleaning basket, while the small parts are unable to pass through the screen. However, since the abrasive material flows readily through the screen, little agitation of the abrasive material and the small parts occurs during the cleaning process unless the person holding the basket shakes the basket with the small parts in it. This is difficult to do while directing the abrasive cleaning material into the open upper end of the basket. Even then, only minimal agitation of the abrasive cleaning material occurs before it passes through the screen. Consequently, even with this basket, cleaning of small parts is difficult and time consuming.